Half-order reactions

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first-order, whereas a second-order reaction would have an apparent order 3/2 for diffusion-limited conditions. 

Let us next look at the rate. From Eq. 18.28 we have for a half order reaction... 

Relative sizes of reactors based on the two models are given in Figure 17.2 for second- and half-order reactions at several conversions. For first order reactions the ratio is unity. At small values of the parameter n and high conversions, the spread in reactor sizes is very large. In many packed bed operations, however, with proper initial distribution and redistribution the value of the parameter n is of the order of 20 or so, and the corresponding spread in reactor sizes is modest near conversions of about 90%. In such cases the larger predicted vessel size can be selected without undue economic hardship. 

Figure 17.2. Relative volumes of maximum-mixed and segregated flow reactors with the same RTDs identified by n = 1 /< , as a function of conversion for second- and half-order reactions. For first-order reactions the ratio is unity throughout.

A zero-order reaction thus becomes a half-order reaction, a first-order reaction remains first order, whereas a second-order reaction has an apparent order of 3/2 when strongly influenced by diffusional effects. Because k and n are modified in the diffusion controlled region then, if the rate of the overall process is estimated by multiplying the chemical reaction rate by the effectiveness factor (as in equation 3.8), it is imperative to know the true rate of chemical reaction uninfluenced by diffusion effects. 

Using concentrations expressed in molarity and time in seconds, what are the units of the rate constant, k, for (a) a zero-order reaction (b) a first-order reaction (c) a second-order reaction (d) a third-order reaction (e) a half-order reaction ... 

Figures 3a and 3b show the variation in dimensionless concentration in the catalyst slab as a function of Thiele modulus for second order and half order reactions respectively. In the case of these reaction orders there is no analytical solution to the...

Figure 3. Variation of concentration in porous catalyst slab for (a) a second order reaction =1,2, 5, 10 and (b) a half order reaction = 0.5, 1.0, 2.0.

Problem 3 What is a half order reaction Derive rate equation for it and discuss its characteristic. 

A reaction in which the reaction rate depends on half power of concentration of a reactant is known as half order reaction, i.e.,... 

The other alkali metals have been less extensively studied. The propagation rates of polystyrylsodium, -potassium, -rubidium and -cesium have been measured in benzene and cyclohexane [72, 73]. The sodium compound still shows half order kinetics in active centre concentration and is presumably associated to dimers. The rates for the rubidium and cesium compounds are directly proportional to the concentrations of the active chains which are presumably unassociated in solution. Absolute kp values can be determined from the propagation rate in this case. Poly-styrylpotassium shows intermediate behaviour (Fig. 11), the reaction order being close to unity at a concentration of the potassium compound near 5 x 10 M and close to one half at concentrations around 10" M. It could be shown by viscosity measurements that association was absent in the low concentration range. In this system both K2 and kp can be measured. The results are summarized in Table 2. The half order reactions show a large increase in kpK between lithium and potassium which... 

Repeat the analysis assuming a half-order reaction, and compare the fit to the data. 

Similar comparisons could be presented for second-order reactions, but several tables or plots would be needed, because most such reactions involve two reactants, with one fed in excess. The ratio of reaction times is higher than for a first-order reaction, but not much higher when there is a large excess of one reactant. For half-order reactions, there is less change in rate with conversion, and the ratio of reaction times is less than for first-order kinetics. 

One concludes that a zero-order reaction in a biofilm either becomes a bulk zero-order reaction or a half-order reaction—depending on the characteristics of the biofilm (/cgf, S, D ff) and on the S concentration at the L S interface. The half-order reaction (see Equ. 4.87) with... 

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